Analytic genericity of diffusing orbits in a priori unstable Hamiltonian systems

TL;DR

This paper proves that Arnold diffusion occurs generically in a class of a priori unstable Hamiltonian systems with analytic perturbations, even without convexity or twist conditions, using geometric methods.

Contribution

It establishes the genericity of Arnold diffusion in non-convex, non-twist Hamiltonian systems with analytic perturbations, expanding understanding of diffusion phenomena.

Findings

Arnold diffusion occurs for generic analytic perturbations.

The set of admissible perturbations is dense and open in the analytic topology.

The perturbative technique applies in various smoothness topologies, including analytic.

Abstract

The genericity of Arnold diffusion in the analytic category is an open problem. In this paper, we study this problem in the following a priori unstable Hamiltonian system with a time-periodic perturbation \[\mathcal{H}_\varepsilon(p,q,I,\varphi,t)=h(I)+\sum_{i=1}^n\pm \left(\frac{1}{2}p_i^2+V_i(q_i)\right)+\varepsilon H_1(p,q,I,\varphi, t), \] where $(p,q)\in \mathbb{R}^n\times\mathbb{T}^n$, $(I,\varphi)\in\mathbb{R}^d\times\mathbb{T}^d$ with $n, d\geq 1$, $V_i$ are Morse potentials, and $\varepsilon$ is a small non-zero parameter. The unperturbed Hamiltonian is not necessarily convex, and the induced inner dynamics does not need to satisfy a twist condition. Using geometric methods we prove that Arnold diffusion occurs for generic analytic perturbations $H_1$. Indeed, the set of admissible $H_1$ is $C^\omega$ dense and $C^3$ open (a fortiori, $C^\omega$ open). Our perturbative technique for the genericity is valid in the $C^k$ topology for all $k\in [3,\infty)\cup\{\infty, \omega\}$.